Liquidity, Innovation, and EndogenousGrowth∗

Semyon Malamud† Francesca Zucchi‡

October 9, 2016

Abstract

We study how financing frictions and corporate cash hoarding affect firms’ invest-

ment in innovation and shape economic growth. By nesting a dynamic corporate

finance model with financing frictions into a model of endogenous growth, we show

that financing frictions have two offsetting effects on growth. First, financing fric-

tions exacerbate entry barriers and deter new firms from innovation (the hurdle

effect). Second, financing frictions spur innovation by incumbents by reducing exit

threats and prompting a substitution from production to innovation (the haven ef-

fect). Financing frictions can foster growth if the haven effect prevails, which is the

case if entry barriers are sufficiently large.

Keywords: Innovation; Cash management; Financing frictions; Endogenous growth

JEL Classification Numbers: G31; G32; O31; O40

∗We thank Jean-Paul Décamps, Michael Fishman, Teodor Godina, Mikhail Golosov, Sebastian Gry-glewicz, Luigi Guiso, Barney Hartman-Glaser, Zhiguo He, Dalida Kadyrzhanova, Leonid Kogan, ArvindKrishnamurthy, Ernst Maug, Konstantin Milbradt, Erwan Morellec, Boris Nikolov, Dino Palazzo, PaolaSapienza, Lukas Schmid, Toni Whited, an anonymous referee, and the participants of the 12th AnnualConference in Financial Economics Research (IDC Herzliya), the 7th EBC Conference (Tilburg), the2016 AFA Annual Meeting (San Francisco), the 2016 EFA Annual Meeting (Oslo), and the 2016 NorthAmerican Summer Meeting of the Econometric Society (Philadelphia) for helpful comments. Any remain-ing errors are our own. Semyon Malamud gratefully acknowledges the Lamfalussy Fellowship Programsponsored by the European Central Bank. The views expressed in the paper are those of the authorsand do not necessarily represent those of the ECB, the Eurosystem, the Federal Reserve System, or theirstaff.†Swiss Finance Institute, EPFL, and CEPR. E-mail: semyon.malamud@epfl.ch‡Federal Reserve Board of Governors. E-mail: francesca.zucchi@frb.gov

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1 Introduction

Innovation is pivotal to economic growth. The emergence of new products and improve-

ments in existing goods shape the world in which we live and drive the dynamics of the

economy. Yet, investment in innovation (or research and development, R&D) is costly

and difficult to finance. It requires a long gestation period before becoming productive,

it is not pledgeable, and its outcome is uncertain. To safeguard financial flexibility, in-

novative firms hoard cash reserves (or liquidity).1 While much work has been done to

understand how financing frictions affect corporate cash management and investment,

there is no work studying how financing frictions jointly affect firms’ production and

innovation decisions and factor into economic growth. This paper seeks to fill this gap.

To this end, we develop a tractable model of the effects of financing frictions and

corporate cash hoarding on innovation and economic growth. The model consists of two

building blocks. The first is a dynamic corporate finance model with financing frictions

for a continuum of incumbents and entrants. We use the model to investigate how con-

strained incumbents manage their production-innovation mix as well as to study entrants’

incentives to invest in innovation. The second building block is a model of endogenous

growth, in which we nest our firm maximization problem. Similar to extant growth mod-

els, innovations by incumbents and entrants that increase the quality of products are the

key drivers of economic growth. We depart from these contributions by recognizing that

innovative firms face financing frictions and, thus, hoard precautionary cash reserves.

We start by solving the intertwined optimization problems of constrained incumbents

and entrants. Each incumbent produces one good and invests in innovation to improve

the quality of this good. Incumbents are financially constrained in that they have costly

access to external financing whenever they seek to refinance current operations or to

market technological breakthroughs. To maintain financial flexibility, incumbents have

an incentive to retain earnings in cash reserves and adjust production and innovation rates

in response to operating shocks. Each incumbent faces the threat of creative destruction—

1 R&D is a major determinant of corporate cash reserves, as illustrated by Lyandres and Palazzo(2016), Ma, Mello, and Wu (2013), Falato, Kadyrzhanova, and Sim (2013), Falato and Sim (2014), andBegenau and Palazzo (2016).

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i.e., losing its market position when an entrant launches a better product in the same

product line. Entrants only invest in innovation and do not produce any goods. To single

out the effects of financing frictions, we compare the optimal policies in the constrained

economy with those associated with an identical, but unconstrained economy.

We show that constrained incumbents scale down production and increase markups

in response to negative operating shocks. Negative operating shocks drain corporate cash

reserves and weaken a firm’s financial strength. In an attempt to avert a costly refinancing,

an incumbent reduces its production rate to limit cash flow risk and cut operating costs.

Our model predicts that incumbents produce less and charge higher markups in the

constrained economy than in the unconstrained economy. A constrained incumbent’s

production rate increases with cash reserves and equals the production rate associated

with the unconstrained economy only at its target cash level. The resulting production

rate is cyclical (and markups countercyclical) to operating (firm-specific) shocks.

A key result of our model is that incumbents may invest more in innovation in the

constrained economy than in the unconstrained economy. Despite the tendency to scale

down production, constrained incumbents may increase their innovation rate in response

to negative operating shocks, thus substituting production for innovation. The reason is

the following. Both the marginal gain and cost of investing in innovation increase as cash

reserves shrink. The marginal gain is the probability-weighted change in firm value when

attaining a technological breakthrough, in which case the firm accesses fresh monopoly

rents and raises funds in light of a success (the achievement of a breakthrough) rather

than a failure (running out of funds because of bad operating performance). The marginal

cost of innovation is the associated decrease in cash reserves, which makes the firm more

constrained. If the gain rises faster than the cost as cash reserves decrease, the optimal

innovation rate decreases with cash reserves. We show that this is the case for constrained

firms with volatile cash flows and low operating margins. For these firms, it is less costly

(in terms of foregone profits) to reallocate resources from production to innovation.

Although this result may be surprising in light of the documented positive relation

between R&D and corporate cash hoarding (see footnote 1), one should not confuse the

relation between R&D and target cash reserves with the relation between R&D and de-

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viations from this target. While extant works have focused on the first relation, our

contribution is to shed light on the second one. Our results help rationalize the observa-

tion that some firms have actually increased their R&D expenditures despite the tighter

constraints during the recent crisis (see Archibugi, Filippetti, and Frenz, 2013). Moreover,

we illustrate which firm characteristics should prompt this behavior.

Our model also shows that constrained entrants spend more in innovation than in the

unconstrained economy if their cash reserves are sufficiently large. Similar to incumbents,

the entrants’ marginal cost and gain from innovation increase as cash reserves decrease.

Differently, entrants do not produce any goods and lack regular cash flows that replenish

cash reserves and smooth R&D expenditures. Thus, the marginal value of entrants’ cash

is larger than that of incumbents, and the increase in the marginal cost of innovation

associated with decreasing cash reserves is more likely to offset the rising gain. When

cash reserves are large (i.e., after a financing round), the marginal cost of innovation

is relatively low, but the marginal gain is greater than in the unconstrained economy.

Conversely, when cash reserves are small, the marginal cost of innovation is large and

causes constrained entrants to innovate less than in the unconstrained economy.

More generally, our model predicts that financing frictions do not need to imply lower

firm values.2 Financing frictions inflate entry costs and cause a decrease in the measure

of active entrants, which lessens incumbents’ exit threats. As a result, incumbents have

longer horizons, are more valuable, and innovate more in the constrained economy. The

increase in incumbent value, in turn, boosts entrants’ incentives to innovate (and to

become the next incumbent). In equilibrium, financing frictions result in fewer entrants,

which can be more valuable though. That is, we show that financing frictions mostly affect

the extensive (rather than intensive) innovation margin, in line with Nanda and Rhodes-

Kropf (2013). Overall, we illustrates that financing frictions importantly affect industry

composition via incentives to innovate, providing theoretical grounds to empirical studies

relating access to financing and composition of aggregate R&D investment.3

2Relatedly, Stangler (2009) finds that cold markets lead to the funding of more successful companies.Anecdotal evidence also suggests that hot periods in innovation financing are associated with less valuablefirms being financed (see Gupta, 2000).

3See, e.g., Brown et al. (2009); Brown et al. (2013); Gao, Hsu, and Li (2014); Kerr and Nanda (2009).

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We next analyze how these firm dynamics affect economic growth. We focus on a

“balanced growth path” equilibrium featuring a constant, endogenous growth rate. In-

cumbents’ and entrants’ contributions to growth are obtained by aggregating their inno-

vation rates via the distribution of cash, which assigns a probability weight to innovation

rates associated with different levels of cash reserves. Thus, the shape of entrants’ and

incumbents’ distribution of cash determines the equilibrium impact of firms’ policies.

Via this “composition” of firms’ policies in equilibrium, financing frictions have two

offsetting effects on growth. First, financing frictions decrease the entrants’ contribution

to economic growth. In fact, although entrants with large cash reserves invest more in

innovation in the constrained economy than in the unconstrained economy, their aggregate

impact is modest because there is a large mass of entrants holding small levels of cash

reserves. This is what we call the “hurdle” effect of financing frictions on growth, which

is growth-decreasing. Second, incumbents increase their investment in innovation as they

expect their profits to last longer before losing their market position due to creative

destruction. This is what we call the “haven” effect of financing frictions on growth,

which is growth-enhancing. Which of the two effects dominates depends on the relative

share of entrants’ and incumbents’ contributions to growth, which is itself endogenous.

We find that the haven effect overtakes the hurdle effect when the entrants’ contribution

to growth is relatively small, which is the case when setup costs are sufficiently large.

Two key predictions follow. First, financing frictions importantly affect the compo-

sition of growth and place entrants at a disadvantage. This finding is consistent with

Brown et al. (2009) and Nanda and Nicholas (2014), who show that tighter constraints

are mostly detrimental for R&D of young firms. Second, financing frictions do not need

to be detrimental to growth. They can be growth-enhancing if the haven effect overtakes

the hurdle effect.

Related literature Our paper contributes to the literature studying cash management

models, which includes Riddick and Whited (2009); Décamps, Mariotti, Rochet, and Vil-

leneuve (2011); Bolton, Chen, and Wang (2011, 2013); Hugonnier, Malamud, and Morellec

(2015); and Décamps, Gryglewicz, Morellec, and Villeneuve (2016). In this strand, the

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papers analyzing the relation between cash reserves and investment consider neoclassi-

cal models of investment and capital accumulation, either incremental (as in Bolton et

al., 2011, 2013) or lumpy (as in Hugonnier et al., 2015). To the best of our knowledge,

our paper is the first to study cash management in a Schumpeterian framework.4 This

problem is economically important given the documented relations between cash reserves

and R&D and between R&D and economic growth (e.g., Caballero and Jaffe, 1993; Ak-

cigit and Kerr, 2016; Kogan et al., 2016), and is not trivial because it involves solving a

problem with an infinite number of (sequentially arriving) stochastic growth options.

The paper also contributes to the literature on innovation financing. Lyandres and

Palazzo (2016) and Ma, Mello, and Wu (2014) investigate and test the relation among

competition, R&D, and cash reserves. Falato, Kadyrzhanova, and Sim (2013) focus on

the relation between investment in intangible capital and corporate cash in a neoclassical

model with no growth. We contribute to this strand by studying cash, production, and

innovation decisions altogether, as well as their ensuing impact on economic growth. Our

model studies the relation between R&D and target cash reserves and between R&D

and deviations from this target. While acknowledging that R&D prompts firms to hoard

precautionary cash, we characterize how firms will adjust their investment in innovation

when operating shortfalls erode cash reserves below their target level.

The link between innovation and corporate cash is supported by vast empirical evi-

dence. Hall (2005) and Hall and Lerner (2010) document that innovation is best financed

through internal funds because it is subject to asymmetric information, it is not pledge-

able, and it has uncertain returns. Brown, Fazzari, and Petersen (2009) document that

innovation decisions are related to the supply of internal and external equity. They re-

port that young publicly traded firms in high-tech industries finance R&D almost entirely

with internal and external equity, as debt financing is difficult due to non-pledgeability.5

Consistently, Hall (2005); Hall and Lerner (2010); Rajan (2012); Brown, Martisson, and

4In neoclassical capital accumulation models, investment implies an immediate increase in output. InSchumpeterian models, the payoff from R&D investment arrives at uncertain (Poisson) times.

5Citing from Brown, Fazzari, and Petersen (2009), “Our results suggest that more attention shouldbe given to equity finance [...] for models that emphasize innovation. [...] External equity is the morerelevant substitute for internal cash flow for young high-tech firms.”

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Petersen (2013); and Acharya and Xu (2016) emphasize the relative importance of equity

rather than debt for the financing of innovation, as we do in our model.

Schumpeter (1942) underlined the link between innovation and economic growth by

introducing the concept of “creative destruction,” which spurred the development of a

subfield of macroeconomics known as Schumpeterian models of endogenous growth (for

a survey, see Aghion, Akcigit, and Howitt, 2014). We nest in this literature by assuming

that growth is spurred by endogenous technological change (see Romer, 1990; Grossman

and Helpman, 1991; or Klette and Kortum, 2004) and that both incumbents and en-

trants invest in innovation (as in Acemoglu and Cao, 2015; Akcigit and Kerr, 2016; and

Acemoglu, Akcigit, Bloom, and Kerr, 2013). Our paper also relates to the literature

studying the effects of financial constraints on economic growth; see Levine (2005) and

Beck (2012). Thus far, however, little attention has been paid to the role of corporate

cash in providing flexibility to innovative firms. Our model seeks to fill this gap.

The paper is organized as follows. Section 2 describes the model. Section 3 solves

the model in the unconstrained economy, which serves as a benchmark to single out the

effects of financing frictions on innovation and growth. Section 4 solves the model in

the constrained economy. Section 5 provides a numerical implementation of the model.

Section 6 concludes. Technical developments are in the Appendix.

2 The model

Time is continuous and uncertainty is modeled by a probability space, (Ω,F,P), equipped

with a filtration, (Ft)t≥0, that represents the information available at time t. We study

an economy in which the representative household maximizes discounted inter-temporal

utility with constant relative risk aversion (CRRA) preferences:

E

[∫ ∞0

e−ρtC1−θt − 1

1− θ

]dt . (1)

In this expression, Ct is consumption at time t, ρ is the discount rate, and θ is the inverse

of elasticity of intertemporal substitution. The household supplies one unit of labor at

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any time and receives a competitive wage. Population size is constant, normalized to one.

2.1 Consumption good sector

There is a unique consumption good, whose output is denoted by Yt. The consumption

good serves as the numeraire of the economy. The consumption good is produced com-

petitively using labor and a multiplicity (a continuum) of inputs, which are indexed by j.

As in Grossman and Helpman (1991), the continuum of inputs has measure one, j ∈ [0, 1].

The production technology of the consumption good is

Yt =1

1− β

∫ 10

X̄1−βjt qβjt dj , β ∈ (0, 1), (2)

where X̄jt denotes the quantity and qjt the quality of input j at time t. Improvements

in the quality of inputs stem from innovation. In the tradition of Schumpeterian growth

models, innovation advances the technological frontier and spurs economic growth via

creative destruction—i.e., new inputs drive the old ones out of the market. Thus, only

the frontier (highest-quality) version of each input is used by the consumption good sector.

2.2 Input sector

There are two types of firms in the input sector: (1) incumbent firms, which actively

produce the frontier inputs and invest in innovation to further enhance their quality; and

(2) entrant firms, which only invest in innovation.

Incumbents The latest innovator of industry j enforces a patent on the frontier version

of input j and becomes the incumbent monopolist of that industry. While producing input

j, the incumbent keeps investing in innovation. Patents last forever but do not prevent

firms from innovating and further improving the quality of inputs.

We adopt the modeling of innovation that is standard in the growth literature, as

surveyed by Aghion, Akcigit, and Howitt (2014). We denote by zjt the innovation intensity

(or innovation rate) of incumbent j at time t. Innovation is costly, and its outcome is

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uncertain. Specifically, an incumbent bearing the flow cost

Φ(zjt, qjt) = ζz2jt2qjt, ζ > 0 (3)

increases the quality of its input at Poisson rate φzjt. These Poisson events represent

technological breakthroughs, which are more frequent if zjt is larger. When the incumbent

of industry j attains a breakthrough, the quality of the input jumps from qjt− to

qjt = λqjt− ,

where λ > 1 represents the size of the quality improvement.6

Taking the demand schedule of the consumption good sector, each incumbent opti-

mally sets the production rate and the associated selling price, which we denote respec-

tively by X̄jt and pjt. The dynamics of incumbents’ cash flows (associated with these

endogenous choices) satisfy

dΠjt =[X̄jt(pjt − 1)− Φ(zjt, qjt)

]dt + σX̄jtdZjt , (4)

where Zjt is a standard Brownian motion representing operating shocks. Operating shocks

are firm-specific and independent across firms. To ease the notation, the marginal cost of

production is normalized to one.

The cash flow process (4) implies that incumbents can make operating profits and

losses. If external financing was costless, losses could be covered with fresh funds whenever

needed. We depart from this assumption and assume that firms face financing costs. As in

previous cash management models, financing costs are modeled in a reduced-form fashion

to capture limited enforcement, asymmetric information, or limited pledgeability.

Specifically, incumbents can raise external funds to cover operating shortfalls and

the flow cost of R&D. This “routine” financing is incurred by the firm during the “dis-

covery” phase of R&D and entails a proportional cost � ∈ [0, 1] and a fixed cost ωqjt.6As in Aghion, Howitt, and Mayer-Foulkes (2005), Acemoglu, Aghion, and Zilibotti (2006), and

Acemoglu and Cao (2015), we abstract from skilled labor in the inputs sector.

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The fixed cost scales with input quality, which directly affects firm size in our model.

The assumption of scalability of the fixed cost with firm size is shared with extant cash

management models (e.g., Bolton, Chen, and Wang, 2011, 2013; Décamps, Gryglewicz,

Morellec, and Villeneuve, 2016).7 Incumbents can also raise funds when they attain tech-

nological breakthroughs—i.e., at the “marketing” phase of R&D—by ceding a surplus

share α ∈ [0, 1] to financiers.8

To maintain financial flexibility, incumbents retain earnings in cash reserves. We

denote cash reserves at time t by C̄jt. Incumbents’ cash reserves earn a rate of return δ,

which is lower than the market interest rate r. As previous cash management models,

we interpret the difference r − δ (the cost of holding cash) as an agency cost of free cash

flows. Alternatively, this cost can be interpreted as a liquidity premium related to the

scarce supply of liquid assets in the economy (see Krishnamurthy and Vissing-Jorgensen,

2012). The dynamics of cash reserves satisfy

dC̄jt = δC̄jt + dΠjt − dD̄jt + dF̄jt + dF̄ Ijt − dΩ̄jt. (5)

In this equation, D̄jt, F̄jt, F̄Ijt, and Ω̄jt are non-decreasing processes that respectively

represent cumulative payouts, cumulative financing obtained during the discovery and

the marketing phases, and cumulative issuance costs. Equation (5) is an accounting

identity illustrating that cash reserves increase with the interests on cash and cash flows

(the first and second terms) and financing (the fourth and fifth terms) and decrease with

payouts (the third term) and issuance costs (the last term).

Each incumbent sets production, innovation, financing, and payout policies to maxi-

mize future net dividends subject to the budget constraint (5) and C̄jt ≥ 0. We denote

incumbents’ value by V (t, C̄j, qj), which is a function of cash reserves and quality at time

t. When creative destruction hits industry j, the associated incumbent exits. The ex-

7While keeping the analysis tractable, this assumption is motivated by the observation that “negativeincentive effects of a more diluted ownership may have costs that are proportional to firm size,” assuggested by Bolton et al. (2011). See also Décamps et al. (2016) and the discussion therein.

8Nash bargaining over surplus between the firm and financiers provides a micro-foundation for ourassumption. Denoting the bargaining power of financiers by α ∈ [0, 1] and the surplus created by S, therents extracted by financiers are Γ∗ = arg maxΓ≥0 Γ

α [S − Γ]1−α = αS, whereas the firm retains (1−α)S.

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iting incumbent entirely recovers its cash reserves but only some fractions ψ and ψE of

the values of mature productive assets (denoted by KTjt) and R&D assets (denoted by

KEjt). We assume that these values increase in quality and are given by KTjt = κT qjt and

KEjt = κEqjt for some κT , κE > 0. Because R&D assets are less tangible than productive

assets, they are more costly to liquidate and, thus, ψ > ψE. Because liquidation is costly,

incumbents never choose to liquidate the firm when cash reserves are positive.

Entrants A mass of penniless entrepreneurs seeks to obtain financing to buy the R&D

technology and start investing in innovation. Entrepreneurs who obtain financing become

active entrants. The mass of active entrants is endogenous and is denoted by mE.

Entrants differ from incumbents in three respects. First, entrants do not produce any

inputs and only invest in innovation (recall that frontier inputs are patent protected).

Second, entrants are more financially constrained than incumbents, as documented by

Nanda and Rhodes-Kropf (2016), among others. Third, entrants’ innovations have wider

breadth than incumbents’ innovations, which results in ex-ante uncertainty regarding the

input that an entrant may improve. This uncertainty dissolves when the entrant targets

an industry after achieving a breakthrough. As a result, the entrants’ cost of innovation

is a function of the average quality of frontier inputs, as we will further explain. Because

entrants’ characteristics are not input-specific, we describe a “representative” entrant.

Active entrants choose their optimal innovation rate, zEt. Similar to incumbents,

zEt regulates the Poisson rate of technological breakthroughs, which is given by φEzEt,

φE > 0. Technological breakthroughs by entrants lead to jumps in quality by a factor

Λ > 1. Sustaining the innovation rate zEt entails a flow cost

ΦE(zEt, q̄t) = ζEz2Et2q̄t.

While this expression has a structure similar to (3), it differs along two dimensions. First,

this cost is proportional to the average quality of frontier inputs

q̄t =

∫ 10

qjtdj,

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rather than to the quality of a specific input j. Second, ζE may differ from ζ.9

We model entrants’ financing frictions following entrepreneurial finance schemes, ac-

cording to which financiers tend to stage their investments, learn about the firm’s po-

tential, and deny further financing if no milestone or breakthroughs are attained (see

Gompers, 1995; Lerner et al., 2012; and Nanda and Rhodes-Kropf, 2016). To keep the

analysis tractable, we model this financing scheme as follows. Consider an active en-

trant that has obtained initial financing to set up the R&D technology, whose cost is

proportional to average quality and is given by K̄Et = κE q̄t. At the outset, the entrant

faces proportional and fixed financing costs denoted respectively by �E and ωE q̄t, as in-

cumbents. These costs increase if the entrant seeks to raise more funds without having

attained a breakthrough. In these cases, successive financing rounds can be so expensive

that the unsuccessful entrant may prefer to liquidate rather than continue. For simplicity,

we assume that this happens at the second financing round (in Appendix A.3, we micro-

found this assumption via learning about an entrant’s type).10 Conversely, technological

breakthroughs are milestones that wane adverse selection. In these cases, entrants can

raise funds by promising a fraction αE ≥ α of surplus to financiers, as incumbents do.

Because external financing is costly, entrants raise some slack at the outset. This

slack serves to finance the flow cost of R&D and is stored as cash reserves. We denote

by C̄Et an entrant’s cash reserves at time t. We assume that the rate of return on cash

is lower for entrants than for incumbents; for simplicity, we set this rate to zero. This

assumption can be motivated by a harsher free cash flow problem caused by entrants’

less tangible assets, which is consistent with Gompers (1995). If we interpret the cost of

cash as a liquidity premium, the lower return on entrants’ cash stems from their need to

keep cash in vaults (rather than investing in money-like securities with higher returns)

because entrants have no cash flows to smooth the flow cost of R&D. Thus, the dynamics

9We do not impose a priori restrictions on the relations between λ and Λ, φ and φE , and ζ and ζE .10In Appendix A.3, we assume that there are two entrant types: good and bad. Good entrants

eventually attain breakthroughs, while bad entrants never do so. Financiers start with a prior probabilityabout an entrant’s type, which they update using Bayes rule. Conditional on observing no breakthroughs,the posterior estimate of the probability of the entrant being of good type is monotonically decreasing overtime. It is then straightforward to show that there is a deterministic time t∗ after which an unsuccessfulentrant exits. When the prior is sufficiently small, only one round of financing is optimal, in agreementwith our assumption.

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of an entrant’s cash reserves satisfy

dC̄Et = −ΦE(zEt, q̄t)dt (6)

until the entrant attains a technological breakthrough or it liquidates after depleting cash

reserves and being denied further financing. As in practice, not all active entrants succeed

(see, e.g., Ewens and Rhodes-Kropf, 2015).

We denote entrant value by VE(t, C̄E, q̄), which is a function of cash reserves and of the

average quality of frontier inputs at time t. Each entrant maximizes its value by setting

innovation, financing, and cash management decisions until it attains a breakthrough

or exits. If it attains a breakthrough, the entrant improves the quality of the input in

industry j from qjt− to qjt = Λqjt− and needs to pay a cost KTjt = κT qjt to set up the

productive assets. If it exits, the entrant recovers a fraction ψE of the investment in R&D

assets, K̄Et. Just like incumbents, entrants never choose to stop operations when holding

positive cash balances because liquidation is costly.

2.3 Balanced growth path

We focus on a balanced growth path equilibrium in which aggregate quantities grow at

the constant rate g, which is endogenously determined. Competition among entrants de-

termines the measure of active entrants, which is pinned down by the free entry condition

VE(t, C̃Et, q̄t;mE) = (1 + �E)(C̃Et + K̄Et) + ωEt .

In this equation, K̄Et + C̃Et is the amount raised at entry (setup cost and slack) and

�E(C̃Et + K̄Et) + ωEt is the associated financing fees.11 The measure of entrants and the

entrants’ innovation rates together define the equilibrium rate of creative destruction.

To solve the model, we first derive incumbents’ and entrants’ optimal policies by taking

the market interest rate and the growth rate as given. We then aggregate firms’ decisions

11Note that in equilibrium, C̃Et depends on mE , the mass of entrants. We consciously suppress thedependence of endogenous quantities on mE to keep the notation simple.

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and derive these equilibrium quantities. An equilibrium is an allocation such that: (i)

incumbents set production, innovation, cash retention and payout, and financing decisions

to maximize their value; (ii) entrants choose innovation, financing, and cash policies to

maximize their value; (iii) the measure of entrants makes the free entry condition binding;

(iv) the consumption good sector maximizes profits; (v) the representative household

maximizes utility from consumption; and (vi) all markets clear.

2.4 Discussion: Corporate finance meets growth

We nest a dynamic corporate finance model with financing frictions into a model of

endogenous growth. Our growth model has a Schumpeterian nature, as innovation is

the engine of growth and new inputs replace the old ones—i.e., growth involves creative

destruction. As we clarify in this section, our key assumptions are borrowed from the

extant literature (see Aghion et al., 2014, for an excellent survey).

Specifically, the innovation technology is drawn from the theoretical industrial orga-

nization and patent race literature (Tirole, 1998). Firms choose their innovation rates,

which affect the Poisson rate of occurrence of technological breakthrough—i.e., larger

innovation rates increase the likelihood of attaining a breakthrough. While early endoge-

nous growth models focus on innovations by new firms only, we contribute to a growing

literature assuming that both incumbents and entrants innovate (e.g., Acemoglu and Cao,

2015; Akcigit and Kerr, 2016), which is motivated by the observation that a large fraction

of R&D in the United States is done by incumbents (see Acemoglu et al., 2013).

Breakthroughs come as improvements in input quality. Following the literature, we

assume that only the frontier version of any input is used in the production of the final

good, so that the latest innovator is the monopolist of the industry. We rule out the case

of limit pricing to keep the analysis tractable and motivate it as previous contributions.

For instance, Aghion and Howitt (1992) assume that innovations are always drastic, so

that a monopolist is unconstrained by potential competition from the previous patents.

Akcigit and Kerr (2016) assume that current and former incumbents of the same product

line enter a two-stage price-bidding game whereby each firm pays a fee to announce its

14

price. Under this assumption, only the new incumbent pays the fee and announces its

price. We adopt similar assumptions in our setup.

Finally, the consumption good is the numeraire of the economy and is produced using

labor and inputs j.12 The interpretation of j as inputs implies that g is the rate of growth

of final good output. If we interpreted j as consumption goods, g would represent the

growth rate of a quality-adjusted consumption index (see Grossman and Helpman, 1991).

3 A benchmark: the unconstrained economy

We first solve the model in an identical, but unconstrained economy (featuring no financ-

ing frictions). In this setting, operating shortfalls can be covered by raising external funds

at no cost. Thus, firms have no incentive to keep cash.

We start by solving the optimization problem of the consumption good sector,

maxX̄j

1

1− β

∫ 10

X̄1−βjt qβjtdj −

∫ 10

pjtX̄jtdj,

which delivers the demand curve for each frontier input: X̄jt =(qβjt/pjt

) 1β. Taking the

demand schedule of the consumption good sector as given, incumbents in the input sector

maximize profits and set the following monopoly price:

pjt =1

1− β≡ p∗. (7)

This price implies a constant markup above the marginal cost of production and is asso-

ciated with the production rate X̄∗jt = qjtX∗, where

X∗ = (1− β)1β (8)

denotes the optimal production rate scaled by quality. Working with scaled quantities

eases the model solution because of their time-invariant nature.

12Because labor is normalized to one and is supplied inelastically, it does not show up in (2).

15

We denote by V (t, q) the value of incumbents in the unconstrained economy (hence-

forth, the subscript j will be suppressed when it causes no confusion). Following standard

arguments, V (t, q) satisfies the Hamilton-Jacobi-Bellman (HJB) equation:

r∗V (t, q)− Vt(t, q) = maxz∗

{βq (1− β)

1β−1 − (z

∗)2

2ζq + φz∗ [V (t, λq)− V (t, q)]

+ x∗d [ψκT q + ψEκEq − V (t, q)]}.

(9)

The terms on the left-hand side represent the return required by investors and the change

in firm value as time elapses, respectively. The first two terms on the right-hand side

represent operating cash flows net of production and innovation costs. The third term

is the probability-weighted change in value when the incumbent markets a technological

breakthrough. The fourth term captures the effect of creative destruction on incumbent’s

value. An incumbent is replaced by a new firm at a rate x∗d, otherwise it preserves its

monopoly power. When creative destruction hits, the incumbent exits and recovers just

a fraction of its investment in productive and R&D assets. The quantities r∗ and x∗d are

endogenously determined later in the analysis.

To solve incumbents’ problem, we conjecture incumbent value to be linear in qt,

V (t, q) = V (qt) = qtv∗ , (10)

for some v∗ > 0 representing scaled (time-invariant) incumbent value. Substituting (10)

into (9) gives an equation that is independent of quality (see equation (37) in Appendix

A.1), which we differentiate to obtain the optimal innovation rate:

z∗ =φ

ζ(λ− 1)v∗ . (11)

The optimal innovation rate increases with firm value v∗, with the Poisson parameter φ,

and with the size improvement λ. Moreover, it decreases with the cost coefficient ζ.

The rate of creative destruction in (9) is determined by aggregating the innovation

rate of active entrants, which we now derive. The value of an active entrant is denoted by

V E(t, q̄), where q̄t represents the average quality of frontier inputs. Following standard

16

arguments, entrant value satisfies the following HJB equation:

r∗V E(t, q̄)− V Et (t, q̄) = maxz∗E

{− (z

∗E)

2

2ζE q̄ + φEz

∗E

[V (t,Λq̄)− κTΛq̄ − V E(t, q̄)

] }. (12)

The terms on the left-hand side represent the return required by the investors and the

change in value as time passes. The first term on the right-hand side represents the flow

cost of R&D. The second term represents the probability-weighted change in value when

the entrant attains a technological breakthrough.13 To solve this maximization problem,

we conjecture that the value of an active entrant is linear in q̄t:

V E(t, q̄) = q̄tv∗E, (13)

where v∗E represents entrant value scaled by average quality. Substituting (13) into equa-

tion (12) yields a time-independent version of the HJB equation (see equation (39)).

Differentiating this HJB equation yields the entrants’ optimal innovation rate:

z∗E =φEζE

max(Λv∗ − v∗E − ΛκT , 0). (14)

The optimal z∗E is strictly positive if the inequality Λv∗ > v∗E + ΛκT holds.

14 Substituting

(14) into the HJB equation, we can solve for v∗E (see Appendix A.1). Combining (13)

with the free entry condition (given by V E(t, q̄;m∗E) = K̄Et in the unconstrained economy)

gives

v∗E(m∗E) = κE.

The free entry condition pins down the equilibrium measure of active entrants, which,

together with z∗E, gives the equilibrium rate of creative destruction:

x∗d = m∗E φE z

∗E = m

∗E

φ2EζE

(Λv∗ − v∗E − ΛκT ) . (15)

13Recall that entrants do not know ex-ante to which industry they will contribute until they target aspecific industry after a breakthrough. When a breakthrough occurs, firm value jumps to V (t,Λq̄t) −κT q̄t =

∫ 10

[V (t,Λqjt)− κTΛqjt] dj. The first term in the integral is the value of the successful entrantafter uncertainty dissolves, whereas the second term represents the setup cost of productive assets.

14If it did not hold, entrants would not have incentives to enter and invest in innovation, and theeconomy would feature incumbents only.

17

In the model, economic growth stems from advancements in the quality of inputs,

which are triggered by the successful innovations of incumbents and entrants. Improve-

ments occur at independent Poisson times in different industries j ∈ [0, 1]. By the law of

large numbers (see Appendix A.1), the rate of economic growth is given by

g∗ = (λ− 1)φz∗ + (Λ− 1)x∗d .

The first (respectively, second) term is the jump in quality due to incumbents’ (entrants’)

breakthroughs times the associated rate of occurrence. Because aggregate consumption

grows at rate g∗ along the balanced growth path, the maximization problem of the rep-

resentative household delivers the market interest rate via the standard Euler equation:

r∗ = ρ+ θg∗.

While the interest rate and the growth rate are the outcome of incumbents’ and entrants’

optimal policies, these rates feed back into corporate decision. Then, equilibrium rates

and corporate policies are determined by a fixed point problem (see (42) in Appendix A.1),

which we solve numerically. In section 5, we analyze the equilibrium of the unconstrained

economy and compare it to the constrained economy, which we derive in the next section.

4 The constrained economy

In the constrained economy, firms face financing frictions and have incentives to hoard

cash reserves. We first study the value-maximizing decisions of incumbents and entrants.

We then derive their stationary distributions of cash, which serve to aggregate firm deci-

sions and derive the equilibrium of the constrained economy.

4.1 Incumbents’ optimal policies

Each incumbent selects retention, payout, financing, production, and innovation policies

to maximize its value. Consider first retention and payout policies. Because precautionary

18

concerns are relaxed when cash reserves are large, the benefit of cash is decreasing in cash

reserves. The cost of cash is constant. We then conjecture the existence of a target cash

level, C̄∗(qt), which trades off costs and benefits (as for the unconstrained economy, we

drop the index j when it causes no confusion). It is optimal to pay out cash exceeding

this target level to shareholders. Below this target level, it is optimal to retain earnings

in cash reserves.

Consider now financing decisions. Because routine financing is costly, incumbents

tap it when cash reserves are depleted. To economize on the fixed cost, it is optimal

to raise a lumpy amount. We denote this amount by C̄∗(qt), which is endogenously

determined. As we show, proportional financing costs imply that C̄∗(qt) is smaller than

C̄∗(qt). Besides this routine financing, incumbents can raise funds when marketing a

higher-quality product after a technological breakthrough, at the cost of ceding a surplus

share to financiers. Incumbents find it optimal to raise an amount that replenishes cash

reserves to C̄∗(λqt). The surplus from this transaction is S(t, C̄, q) = V (t, C̄∗(λq), λq) −

(C̄∗(λq)− C̄)− V (t, C̄, q), but incumbent value only increases by (1− α)S(t, C̄, q).15

Next, consider production and innovation decisions. As in the unconstrained econ-

omy, incumbents choose their optimal production rate by taking the demand schedule of

the consumption good sector as given. Differently, optimal production (and innovation)

decisions depend on the level of cash reserves. Using standard arguments, it possible to

show that firm value satisfies the following HJB equation for c ∈ [0, C̄∗]

rV (t, C̄, q)−Vt(t, C̄, q) = maxz,X̄

{σ22X̄2Vcc(t, C̄, q) +

[δC̄ + (p− 1)X̄ − z

2

2qζ

]Vc(t, C̄, q)

+ φz(1− α)[V (t, C̄∗(λq), λq)− (C̄∗(λq)− C̄)− V (t, C̄, q)

]+ xd[ψκT q + ψEκEq + C̄ − V (t, C̄, q)]

}. (16)

The left-hand side admits an interpretation analogous to (9). The first and second terms

15If an incumbent does not raise funds when marketing a breakthrough, its value jumps from V (t, C̄, q)

to V (t, C̄, λq). If it raises funds, the surplus created (S̃(t, C̄, q) = V (t, C̄∗(λq), λq) − (C̄∗(λq) − C̄) −V (t, C̄, λq)) needs to be shared with financiers. While we assume that financiers receive a fraction α

of S = S̃ + (V (t, C̄, λq) − V (t, C̄, q)) for tractability, this is without loss of generality and it is alwaysoptimal for the firm to raise funds upon a breakthrough, see Appendix A.2.1.

19

on the right-hand side capture the effect of cash flow volatility and cash accumulation.

The third term captures the change in value when the incumbent attains a technological

breakthrough. The fourth term captures the effect of exit due to creative destruction.

To solve the incumbents’ problem, we conjecture that incumbent value scales with qt:

V (t, C̄, q) = qt v

(C̄tqt

)≡ qt v(c) (17)

for some function v(c), which represents scaled (time-independent) firm value. Moreover,

c ≡ C̄t(qt)/qt, C∗ ≡ C̄∗(qt)/qt C∗ ≡ C̄∗(qt)/qt

denote scaled cash reserves, the scaled target cash level, and the scaled amount raised at

(routine) refinancing events, respectively. Optimal policies are determined by solving the

HJB equation for v(c):

maxz,X

{σ22X2v′′(c) + v′(c)

[δc + X1−β −X − z

2

2

ζ

L

](18)

+ (1− α)φz [λ(v(C∗)− C∗)− (v(c)− c)] + xd [ψκT + ψEκE + c− v(c)]− rv(c)}

= 0,

where X(c) ≡ X̄t/Qt represents scaled production. As in extant cash management mod-

els, the marginal value of cash is monotone decreasing in c (that is, v′′(c) ≤ 0). The

maximization problem (18) has an interior solution for any c.

Differentiating (18) with respect to X yields the optimal production rate:

X(c) = F (A(c)), (19)

where F is a decreasing function of A:

A(c) = −σ2v′′(c)

v′(c)(20)

which represents the incumbent’s effective “risk aversion” and depends on the level of cash

reserves (see Appendix A.2.1). Moreover, differentiating (18) with respect to z yields the

20

optimal innovation rate:

z(c) =φ (1− α)

ζ

(λw∗ − v(c) + c)v′(c)

, (21)

where w∗ represents the scaled value of an incumbent holding its target cash level, net

of cash: w∗ ≡ w∗(C∗) = v(C∗) − C∗. The numerator in (21) is a function of c and

represents the change in firm value when the incumbent achieves a breakthrough, which

represents the marginal gain of innovation. The denominator is also a function of c and

represents the marginal loss of using cash to finance innovation, which is the marginal

cost of innovation.

To derive incumbent value and the two thresholds C∗ and C∗, we substitute (19) and

(21) into (18) and solve the resulting differential equation (equation (45)) subject to the

following boundary conditions:

v(0) =v(C∗)− C∗(1 + �)− ω, (22)

v′(C∗) =1 + �, (23)

v′(C∗) =1, (24)

v′′(C∗) =0 . (25)

Condition (22) means that an incumbent raises the amount C∗ whenever cash reserves

are depleted. We only consider parameter values for which the following inequality

v(C∗)− C∗(1 + �)− ω > ψκT + ψEκE

holds, which guarantees that firm value after refinancing (the left-hand side) is preferred

to liquidation (the right-hand side). The optimal issue amount C∗ equalizes the marginal

benefit and cost of financing, such that (23) holds. Condition (24) means that the value

of one dollar at C∗ equals its value if paid out. When cash reserves are close to C∗, the

firm’s effective risk aversion, A(c), goes to zero and precautionary concerns are gradually

relaxed. Above C∗, excess cash is paid out and incumbent value is linear in cash reserves;

21

i.e., v(c) = v(C∗) + c −C∗ for c > C∗. Finally, condition (25) guarantees that the target

cash level maximizes firm value. We summarize our findings in the next proposition.

Proposition 1 The value of an incumbent is given by (17), where v(c) is the unique

concave solution to (18) that satisfies the boundary conditions (22)-(25) (see (45) in

Appendix A.2.1). The value of an incumbent holding its target level of cash reserves is

given by w(C∗) +C∗, where w(C∗) obtains by substituting conditions (24)-(25) into (45):

w(C∗) =v(C∗)− C∗ (26)

=2 [µ∗ + xd(ψκT + ψEκE)− (r − δ)C∗]

xd + r + [(xd + r)2 − 2ϕ2(λ− 1)2[µ∗ + xd(ψκT + ψEκE)− (r − δ)C∗]]1/2

with µ∗ ≡ β (1− β)1β−1 and ϕ2 ≡ φ

2(1−α)2ζ

.

So far, we have taken the rate of creative destruction as given. We now micro-found

it by studying entrants’ maximization problem.

4.2 Entrants’ optimal policies

We now derive entrants’ optimal policies by starting from cash retention. As it is for

incumbents, the marginal benefit of entrants’ cash decreases with cash reserves. Thus,

there should be a target cash level at which the marginal value of cash equals one, mak-

ing it optimal to pay out excess cash. Yet, entrants never reach this target and never

pay out cash. Because entrants do not produce any goods, their cash reserves decrease

monotonically as cumulative R&D expenditures grow (see equation (6)).

Using standard arguments, entrant value satisfies the following HJB equation:16

rV E(t, C̄E, q̄)− V Et (t, C̄E, q̄) = maxzE

{− z

2E

2ζE q̄t V

Ec (t, C̄E, q̄)

+ φEzE(1− αE)[V (t, C̄∗,Λq̄)− κTΛq̄ − (C̄∗(Λq̄)− C̄E(q̄))− V E(t, C̄E, q̄)

] }.

(27)

16Similar to the unconstrained economy, V (t, C∗,Λq̄t)− κTΛq̄ =∫ 1

0[V (t, C∗,Λqjt)− κTΛqjt] dj is the

jump in firm value after a breakthrough. The first term in the integral is firm value when targeting anindustry after a breakthrough, whereas the second one is the setup cost of productive assets.

22

To solve this optimization problem, we scale entrant value and cash reserves by the

average quality of frontier inputs, which we denote respectively by vE(cE) = VE/q̄t and

cE = C̄Et/q̄t. Moreover, we denote by ĈE ≡ C̃Et/q̄t the scaled amount raised at entry.

Substituting these expressions into (27) delivers a time-independent HJB equation (see

equation (47) in Appendix A.2.1), which we differentiate to obtain the optimal innovation

rate:

zE(cE) =φE (1− αE)

ζE

max(Λw∗ − vE(cE) + cE − ΛκT , 0)v′E(cE)

. (28)

Similar to the unconstrained economy, the inequality Λw∗ − vE(cE) + cE − ΛκT > 0

guarantees that technological breakthroughs create value for entrants. To derive entrant

value as well as the threshold ĈE, we substitute (28) into (47) and solve the resulting

differential equation (equation (48)) subject to the following boundary conditions:

v′E(ĈE) = 1 + �E, (29)

vE(ĈE) = (1 + �E)(ĈE + κE) + ωE, (30)

vE(0) = ψEκE (31)

At the outset, entrants raise financing to cover the setup cost (κE) and to have some slack

(ĈE) to fund R&D costs. Condition (29) means that ĈE is optimally set to equate the

marginal benefit (the left-hand side of (29)) and the marginal cost (the right-hand side)

of external financing. Condition (30) represents the free entry condition, which we use to

pin down the equilibrium mass of entrants.17 The left-hand side of (30) is the value of an

incumbent right after entry, whereas the right-hand side is the total cost of entry (setup

cost, cash, and financing fees). Finally, condition (31) implies that when cash reserves

are depleted, entrants recover a fraction ψE of their investment in R&D assets.

Notably, the analysis illustrates that incumbents’ and entrants’ decisions are inter-

twined. Entrants’ innovation rates affect the rate of creative destruction, which deter-

mines incumbents’ exit threats and their incentives to invest in innovation (by affecting

17Because entrant value is concave in cE , the condition Λw∗− vE(cE) + cE −ΛκT ≥ 0 that guarantees

zE(c) > 0 is satisfied for any cE > 0 if it holds at ĈE . By (30), this is the case if Λw∗ ≥ κE + ΛκT +

�E(κE + ĈE) + ωE .

23

w∗ in equation (21)). Any change in incumbents’ value, in turn, affects entrants’ incen-

tives to innovate and to become the next incumbent (note that w∗ also enters equation

(28)). We now delve deeper into the properties of production and innovation decisions.

4.3 Analyzing optimal policies

Production and markups Equation (19) illustrates that the dynamics of X(c) depend

on the firm’s effective risk aversion A(c) via the function F . The function F is monotone

decreasing in A and, hence, so is X(c) (see Appendix A.2.1). Because A(c) decreases with

c, X(c) increases with cash reserves. The intuition behind the math is the following. When

an incumbent’s cash reserves decrease, the associated increase in effective risk aversion

makes it optimal to limit operating risk and scale down production. Because effective

risk aversion is negligible at the target cash level (condition (25) implies A(C∗) = 0), the

production rate in the constrained economy equals the unconstrained production rate (8)

only when c = C∗. As a result, X(c) < X∗ for all c < C∗.

Given the demand schedule of the consumption good sector, selecting quantities is

equivalent to setting prices:

p(c) = X(c)−β ≥ (X∗)−β = 11− β

.

Notably, financing frictions lead incumbents to deviate from the constant price associated

with the unconstrained economy. In the constrained economy, the markup (p(c) − 1)

varies with the level of cash reserves and exceeds the markup that the same firm would

set in the unconstrained economy. In the wake of negative shocks depleting cash reserves,

constrained incumbents decrease production and increase markups. As a result, financial

constraints cause markups to be countercyclical to firm’s operating shocks. Corollary 2

summarizes these results.

Corollary 2 Financing frictions lead incumbents to decrease their production rate and

charge larger markups in comparison to the unconstrained economy. The decrease in the

production rate (and the increase in markups) is greater as cash reserves decrease.

24

Extant models of endogenous growth have stressed that firms’ incentives to invest

in innovation arise from the perspective of earning monopoly rents upon attaining a

breakthrough (e.g., Aghion and Howitt, 1992; Aghion, Akcigit, and Howitt, 2014; Romer,

1990). Corollary 2 warns that financing frictions and liquidity constraints may generate a

distinct, reverse relation, which is novel to the literature. Innovative firms drain their cash

reserves faster, which makes it optimal to decrease their production rate, limit operating

volatility, and charge higher markups.

In our model, production decisions affect the volatility of cash flows, which is a func-

tion of c and given by σX(c). This result has interesting implications for the relation

between cash flow volatility and cash reserves (e.g., Bates, Khale, and Stulz, 2009). As

previous contributions, our model suggests that cash flow volatility affects firm’s cash

management. On top of this relation, our model sheds light on a potential feedback effect

from cash reserves to cash flow volatility via optimal production decisions. After positive

(respectively, negative) operating shocks, cash reserves increase (decrease), the curvature

of the value function decreases (increases), and the firm is willing to take on more (less)

risk. The optimal production rate rises (decreases), and so does cash flow volatility.

Innovation By (21), the optimal innovation rate is a multiple of the ratio (λw∗−v(c)+c)v′(c)

.

As explained, the numerator is the gain from investing in innovation, whereas the de-

nominator represents the marginal “cost” of using cash to finance innovation. Both the

gain and the cost decrease with cash reserves. The gain decreases in c because v′(c) ≥ 1,

which implies that any breakthrough-driven jump in firm value is more valuable when

cash reserves are small. The cost decreases in c because the precautionary benefit of cash

is greater when cash reserves are small. As a result, z(c) decreases with cash reserves

if the gain increases at a higher rate than the cost as cash reserves shrink. A direct

calculation delivers the following result.

Corollary 3 z(c) is decreasing in cash reserves at a given cash level c if the following

inequality holds:v′(c)− 1

λw∗ − v(c) + c≥ σ−2A(c). (32)

25

If two firms have the same fundamental characteristics but different cash reserves, the

firm with smaller cash reserves may invest more in innovation if (32) holds. That is, the

firm may substitute production for innovation to increase the probability of attaining a

technological breakthrough. When a breakthrough occurs, the firm earns monopoly rents

related to the brand-new technology and can raise outside funds in light of a “success”

rather than a “failure” (i.e., running out of funds due to operating losses).

While this result may be surprising in light of the documented positive relation be-

tween R&D and corporate cash (see the references in footnote 1), one should not confuse

the relation between R&D and target cash reserves with the relation between R&D and

deviations from this target. The first relation refers to ex-ante heterogeneity—firms differ

in their R&D technologies (e.g., in the parameters φ, λ ζ), choose different innovation

rates, and set different C∗—whereas the second focuses on ex-post heterogeneity—ex-ante

identical firms (setting the same C∗) have different cash levels because of idiosyncratic

operating shocks. Extant works have focused on the first relation, whereas we shed light

on the second by characterizing how firms adjust their innovation rates when operating

shocks erode their cash reserves below the target level. Although the richness of our

model prevents us from deriving analytical comparative statics, section 5.1 provides a

thorough investigation of which firm characteristics prompt the increase in z(c) as cash

reserves decrease. We find that constrained firms with more volatile profits and lower

margins should increase their innovation rate in the face of decreasing cash reserves.

As it is for incumbents, equation (28) reveals that entrants’ optimal innovation rate

is also a multiple of the gain-cost ratio, (Λw∗−vE(cE)+cE−ΛκT )

v′E(cE). Both the gain and the cost

decrease with cash reserves, as for incumbents. Yet, the level and the rate of decrease

of the cost (i.e., the marginal value of cash) is larger for entrants than for incumbents.

The reason is that cash is more valuable for entrants as it is the only resource available

to fund R&D expenditures. Indeed, entrants do not produce any good and, thus, lack

regular cash flows that can serve to finance R&D and replenish cash reserves. Because

the marginal value of cash is relatively large for entrants, the increase in the marginal

cost of innovation as cash reserves decrease shadows the increase in the gain. Entrants

decrease their innovation rate when cash reserves shrink, in order to deplete cash reserves

26

at a slower pace and have relatively more time to achieve a breakthrough before running

out of funds. While analytical comparative statics are not viable, we study the entrants’

innovation rate across different parameterizations in section 5.1.

4.4 Aggregation

We now aggregate incumbents’ and entrants’ optimal choices. To do so, we derive the

stationary cross-sectional distributions of cash of incumbents and entrants, which we

denote respectively by η(c) ∈ [0, C∗] and ηE(c) ∈ [0, ĈE].

The distributions of cash Our analysis so far shows that constrained firms adapt

their production and innovation choices to their cash positions, which provide firms with

financial flexibility. The distribution of cash allows us to weigh these decisions across the

population of active firms with different levels of cash reserves.

Incumbents’ level of cash reserves varies because of operating profits, losses, and in-

novation expenditures. Moreover, cash reserves are reflected at C∗ because of payouts,

and jump upward at refinancing events. We show that the distribution of this controlled

process (see equation (54)) satisfies the following Kolmogorov forward equation:

1

2

(σ2(c)η(c))

)′′ − (µ(c)η(c))′ − xdη(c) − φz(c)η(c) = 0 (33)with µ(c) = δc+X1−β(c)−X(c)− z

2(c)

2ζ and σ(c) = σX(c).

We interpret this equation heuristically and report the technical details in Appendix

A.2.2. In any time interval, the probability mass can move to adjacent points because

of operating cash flows and volatility (the first two terms). Else, it moves because of

creative destruction (which hits at rate xd) or because of technological breakthroughs

(which occur at rate φz(c)). Equation (33) is solved for a continuous η(c) subject to the

27

following boundary conditions:

η(0) = 0

σ2(0)η′(0) + σ2(C∗)[η′(C∗+)− η′(C∗−)] = 0∫ C∗

0

η(c)dc = 1.

(34)

The first condition implies that there is no probability mass at zero, as incumbents move

to C∗ as soon as they deplete cash reserves. This financing policy implies an inflow of

mass at C∗ and a kink in the distribution, as captured by the second condition (in which

η′(C∗+) 6= η′(C∗−)). The last condition guarantees that the area under the curve is one.

We next turn to the distribution of entrants’ cash. Each entrant’s cash reserves

deterministically decrease over time, until the entrant attains a breakthrough or depletes

cash reserves and liquidates. The exiting firm is immediately replaced by a new entrant,

so that the free entry condition is always binding. The distribution of entrants’ controlled

cash reserves (see equation (58)) satisfies:

−(−z

2E(c)

2ζEηE(cE)

)′− zE(cE)φEηE(cE) = 0 .

In any time interval, the probability mass is driven down by R&D expenditures (the first

term). Moreover, the probability mass moves because of technological breakthroughs,

which happen at rate φEzE(c). To uniquely pin down the distribution, we impose the

unit mass condition:∫ ĈE

0ηE(cE)dcE = 1.

General equilibrium quantities We next use the distributions of cash to derive the

equilibrium quantities of the constrained economy. Along the “balanced growth path,”

all aggregate quantities grow at the endogenous, constant rate g. As it is for the uncon-

strained economy, the law of large numbers implies that the growth path is smooth and

is driven by improvements in quality of inputs (i.e.,∫ 1

0qjtdj = e

gt). In equilibrium, the

28

growth rate is the sum of incumbents’ and entrants’ contributions to growth:

g = (λ− 1)φ∫ C∗

0

z(c)η(c)dc︸ ︷︷ ︸gI

+ (Λ− 1)φEmE∫ ĈE

0

zE(cE)ηE(cE)dcE︸ ︷︷ ︸gE

. (35)

Because innovation decisions are independent and identically distributed across firms, the

law of large numbers implies that the contribution of incumbents to economic growth (the

first term in (35)) is given by the size of their quality improvements multiplied by their

average innovation rate. Similarly, the contribution of entrants (the second term in (35))

is given by the size of their quality improvements times the average innovation rate across

the measure of active entrants, which represents the rate of creative destruction:

xd = φEmE

∫ ĈE0

zE(cE)ηE(cE)dcE.

The measure of active entrants is pinned down by the free entry condition (30).

Equilibrium consumption also grows at the endogenous rate g. The household’s in-

tertemporal choice delivers the standard Euler equation

g =r − ρθ

.

Together with (35), the Euler equation pins down the equilibrium interest rate:18

r = ρ+ θ

[(λ− 1)φ

∫ C∗0

z(c; r)η(c; r)dc+ (Λ− 1)xd(r)]. (36)

Equations (35) and (36) illustrate that while incumbents and entrants set optimal

policies by taking r as given, their policies actually feed back into r. This feedback implies

that any direct increase (or decrease) in g has a “dampened” effect in equilibrium via its

impact on r. As an illustration, consider an exogenous improvement in the incumbents’

R&D technology; e.g., an increase in φ. Incumbents’ value and their innovation rates

increase, which makes entry more attractive and spurs creative destruction. These effects

18We numerically solve this fixed point equation under the technical constraints r > δ and r > g.

29

lead to a direct increase in g. The rise in g leads to a rise in r via the representative

household’s intertemporal problem. The larger r, however, implies a greater cost of

capital for firms, which depresses firm values and innovation rates. This countervailing

effect dampens the direct increase in g. Similarly, any decrease in aggregate innovation

rates leads to a decrease in g, which, in turn, reduces r. Yet, the lower r implies a lower

cost of capital and, thus, larger innovation rates that partly offset the initial decline in g.

In section 5.2, we further analyze the properties of the equilibrium and compare it to

the unconstrained economy.

5 Model analysis

This section provides additional results by comparing the constrained and unconstrained

economies via numerical examples. Table 1 reports the baseline parameterization. We set

λ = 1.04 and Λ = 1.10 to capture the observation that innovations by entrants tend to be

path-breaking, whereas innovations by incumbents tend to be incremental (e.g., Gao et

al., 2014; Acemoglu et al., 2013). Because they are more radical, entrants’ breakthroughs

have longer gestation periods and are more costly than those of incumbents. Hence, we

set φE = 1.5 to be lower than φ = 3 and ζE = 2 to be larger than ζ = 1.

As external funds are more expensive for entrants than for incumbents, we set � = 0.06

and ω = 0.01 for incumbents and �E = 0.10 and ωE = 0.03 for entrants, which are in the

range of the estimates of Hennessy and Whited (2007) and Altinkilic and Hansen (2000).

We set the surplus to financiers when funding a breakthrough equal to α = 0.02 and

αE = 0.07. We set ψ = 0.9, which implies that incumbents lose 10% of their investment

in productive assets at exit (see Hennessy and Whited, 2007). As R&D assets are less

tangible, we set ψE = 0.6. Finally, we set β = 0.16 and σ = 0.40, implying that the cash

flow volatility σX(c) varies between 5% (when c = 0) and 13.8% (when c = C∗).

30

5.1 Production, innovation, and firm values

Figure 1 shows optimal production and innovation decisions as well as firm values. The

top panel shows that the incumbents’ production rate associated with the constrained

economy increases with cash reserves. It is equal to the production rate associated with

the unconstrained economy only when the firm holds its target cash level (at which effec-

tive risk aversion is zero as financial constraints are relaxed), while it is below otherwise.

Cash flow volatility also increases with cash reserves.

In the middle-left panel of figure 1, the incumbents’ innovation rate associated with

the constrained economy is larger than that associated with the unconstrained one, and

more so for small cash reserves (z(c) is 38% larger than z∗ for c close to zero). The

innovation rate decreases with cash reserves when these are sufficiently small, and it is

quite flat otherwise. Recall that this pattern arises if the gain from innovation increases

more than the cost as cash reserves decrease. Comparing the middle-left panel of figure

1 with figure 2 reveals that this is the case for firms with volatile cash flows (high σ) and

low margins (small β). Firms with smaller β choose larger production rates and charge

lower markups, which makes it less costly (in terms of foregone profits) to substitute

production for innovation. Firms with larger σ quickly replenish or deplete their cash

reserves because of their volatile profits. For these firms, the marginal cost of innovation

increases at a lower rate than the gain as cash reserves wane. Notably, if β is sufficiently

large or σ is sufficiently small, z(c) increases with cash reserves.

In figure 3, we vary the magnitude of financing costs. Fixed financing costs make

it optimal to raise funds in lumps, which makes refinancing events more expensive (and

cash more valuable) compared with a setup with only proportional costs. For larger fixed

costs, the increase in the marginal cost of innovation associated with decreasing cash

reserves can locally offset the rising gain, which can cause z(c) to be non-monotonic in

cash when cash reserves are almost depleted (and refinancing is more likely).19 Yet, the

change in z(c) is not very sizable—Brown et al. (2013) consistently find that shifts in

the supply of equity (and the change in financing frictions thereof) have little impact

19As we explain in section 5.2, the aggregate impact of this drop is negligible because the mass ofincumbents with low cash reserves is small (η(0) = 0).

31

on innovation decisions of mature firms. Absent fixed costs, larger financing costs � are

associated with an upward shift in z(c) because the gain from innovation increases at

a larger rate than the cost.20 Differently, a larger α is associated with a lower z(c) as

it trims incumbents’ gain from marketing a breakthrough.21 Interestingly, the financing

friction that discourages incumbents’ innovation is the one incurred at the “marketing”

phase (α) rather than those incurred during the “discovery” phase (�, ω).

Via the threat of creative destruction, incumbents’ innovation choices also depend on

the financing frictions borne by entrants. The middle-right panel of figure 3 shows that

a greater ωE is associated with higher z(c). The reason is that an increase in entrants’

financing frictions reduces the threat of creative destruction for incumbents (as we confirm

in section 5.2). As a result, w∗ in (21) increases and, thus, incumbents have greater

incentives to invest in innovation. Similar arguments imply that z(c) shifts upward as a

result of larger entry costs (see figure 3, bottom-left panel).

Turning to entrants, the bottom-left panel of figure 1 shows that zE(cE) exceeds z∗E

if cash reserves are sufficiently large. The entrants’ innovation rate increases with cash

reserves. As previously explained, cash is more valuable for entrants than for incumbents

because it is the only resource available to finance the flow cost of R&D. When cash

reserves decrease, the increase in the marginal cost of innovation more than offsets the

rising gain and, thus, zE(cE) increases with cE.

Section 5.2 illustrates that greater setup costs or financing frictions reduce the rate

of creative destruction. Yet, comparing the bottom-left panel of figure 1 with figure 4

highlights that a larger (smaller) κE is associated with larger (smaller) zE(cE), especially

for high cash levels. Moreover, more severe financing costs have a modest effect on zE(cE).

The reason is that larger entry costs or financing frictions reduce entrants’ innovation via

the extensive margin—i.e., the measure of active entrants, mE (see figure 6)—rather than

20We do not display this environment in the figure. In a previous version of the paper, we solved forthe case with proportional costs only (see Appendix A.2.1). Absent fixed costs, firms find it optimal toraise an infinitesimal amount whenever cash reserves are depleted. At c = 0, the marginal gain and costof raising funds are equated: v′(0) = 1 + �. Absent fixed costs, the marginal value of cash is relativelylower for any c. The plots associated with this environment are available upon requests.

21For very large values of α, z(c) can be smaller than z∗. Note, however, that in our model werealistically impose that α ≤ αE .

32

the intensive one—i.e., the innovation rate of active entrants, zE(cE). In equilibrium,

fewer entrants increase the value of incumbents as well as the value of becoming the next

incumbent for an active entrant (indeed, w∗ enters both (21) and (28)). As a result,

innovation rates of active entrants can even increase with larger entry hurdles.

The bottom-right panel of figure 4 shows that higher recovery rates of R&D assets are

associated with larger innovation rates for small levels of cash reserves. Larger (lower)

recovery rates mean that liquidation is less (more) costly, which makes entrants more (less)

willing to invest in innovation, consistent with Acharya and Subramaniam (2009). From

the incumbents’ perspective, this positive effect is offset by an increase in the threat of

creative destruction, which in turn reduces incumbents’ incentives to invest in innovation

(see figure 3, bottom-right panel).

Importantly, our analysis shows that financing frictions can lead to an increase in firm

values (figure 1, middle- and bottom-right panels). In the unconstrained economy, entrant

value is equal to v∗E = κE by the free entry condition. In the constrained economy, the

free entry condition accounts for issuance costs and the cash reserves with which entrants

optimally start. The value of constrained entrants ranges from vE(0) = ψEκE < v∗E to

vE(ĈE) > v∗E (see equation (30)), implying that constrained entrants exceed their uncon-

strained value if cash reserves are sufficiently large. When cash reserves are large, entrants

invest more in innovation and have a larger probability of attaining a breakthrough, which

increases their value. When cash reserves are small, conversely, financing and liquidation

costs depress the value of entrants below their unconstrained value. Because the rate of

creative destruction is smaller in the constrained economy (as we discuss in section 5.2),

incumbents invest more in innovation and are more valuable than in the unconstrained

economy. These findings are consistent with Stangler (2009), who finds that cold markets

(i.e., times characterized by tighter financial constraints) lead to funding of more success-

ful firms, as well as with anecdotal evidence suggesting that hot markets are associated

with less valuable firms being financed (see Gupta, 2000).

33

5.2 Financing frictions and growth

Figure 5 shows the distribution of cash of incumbents and entrants. Incumbents’ cash

reserves can grow because of retained earnings and external financing, whereas entrants

build cash reserves only by raising external funds. An entrant’s cash reserves decrease

over time, until it attains a breakthrough or runs out of funds. Entrants decrease their

innovation rate when cash reserves are small, which helps deplete reserves at a slower

pace. The ensuing entrants’ distribution of cash is non-monotonic, and a large mass is

concentrated around cE = 0, in line with the evidence (e.g., Lins, Servaes, and Tufano,

2010).22 Conversely, the incumbents’ cash reserves are relatively large most of the time

as most of the mass is concentrated around large levels of cash reserves.

The shape of the distributions plays a key role in determining how firms’ policies

aggregate in equilibrium. Incumbents’ and entrants’ contributions to growth (respectively,

gI and gE in (35)) are obtained by weighing firms’ innovation rates using the distributions

of cash. This implies that financing frictions affect growth not only through firms’ optimal

innovation rates (which are functions of cash reserves), but also through the probability

with which firms will find themselves with a given level of cash reserves (and, thus,

different degrees of financial flexibility).

In our baseline parameterization, the rate of economic growth is about 2.00% (re-

spectively, 2.55%) in the constrained (unconstrained) economy. Entrants generate 69.8%

of growth in the unconstrained economy, compared with only 54.9% in the constrained

economy.23 The reason is that the entrants’ distribution has a large mass around low

cash levels, for which the constrained entrants’ innovation rate falls below the uncon-

strained rate (see figure 1). Conversely, even if the contribution of incumbents with very

low levels of cash reserves is modest, the incumbents’ contribution to growth is larger

as their innovation rates in the constrained economy exceed those in the unconstrained

economy. Notably, financing frictions importantly change the composition of growth and

place entrants at a disadvantage, consistent with Brown et al. (2009) and Nanda and

22Differently, existing cash management models (e.g., Bolton, Chen, and Wang, 2011) generate adistribution that is monotonically increasing and has most mass concentrated at the target cash level.

23In Acemoglu, Akcigit, Bloom, and Kerr (2013), entrants generate 58% of economic growth.

34

Nicholas (2014).

Figure 6 compares the constrained and unconstrained economies and investigates how

the composition of growth changes as entry costs vary. First, the figure shows that the

entrants’ contribution to growth decreases with the magnitude of entry costs. Larger

entry costs do not decrease the optimal innovation rate of active entrants (as discussed in

section 5.1) but importantly decrease the equilibrium measure of active entrants (third

panel of figure 6). By acting as additional entry barriers, financing frictions lead to a

further reduction in the measure of active entrants and, thus, to a further decrease in the

entrants’ contribution to growth. Nanda and Rhodes-Kropf (2013) consistently find that

shifts in capital supply largely impact the extensive margin of innovation by new firms.

We label this growth-decreasing effect as the hurdle effect of financing frictions on growth

because financing frictions act as entry barriers that prevent innovations by new firms.

Second, figure 6 shows that the incumbents’ contribution to growth increases with

entry costs and is larger in the constrained economy. In fact, the decrease in the entrants’

contribution to growth reduces the incumbents’ probability of being hit by creative de-

struction. The lower threat of creative destruction shifts the incumbents’ optimal inno-

vation rate upward (as shown in figure 3 and discussed in section 5.1). When aggregated

across the population of incumbents via the stationary distribution, this shift translates

into a larger contribution of incumbents to growth. We label this growth-enhancing effect

as the haven effect of financing frictions on growth because financing frictions effectively

shield incumbents from exit threats.

Figure 7 shows the compound effect of these strengths and illustrates that financing

frictions have a non-monotonic impact on growth.24 If κE is low, the hurdle effect domi-

nates and an increase in financing frictions leads to a decrease in growth. In this case, the

decrease in the rate of creative destruction more than offsets the increase in incumbents’

innovation rate. Conversely, if κE is sufficiently large, the haven effect can dominate and

an increase in financing frictions may lead to a rise in g. In this case, the decrease in

the rate of creative destruction is more than offset by the increase in incumbents’ inno-

24In this figure, we plot g up to the critical κE for which xd is zero. Above this level (which variesacross parameterizations), innovation is pursued by incumbents only.

35

vation rates. A key implication of our model is that financing frictions do not need to

be detrimental to growth. Indeed, financing frictions can spur growth if the haven effect

dominates—i.e., when entry barriers are sufficiently large.

6 Conclusion

In the aftermath of the recent financial crisis, financial constraints have raised much

attention among academics and policymakers. While their effects have been mostly in-

vestigated from a financial stability perspective, their impact on economic growth is still

poorly understood. This paper seeks to fill this gap by building a model of endogenous

growth featuring financially constrained firms.

Comparing our constrained economy with an identical but unconstrained economy, we

single out the effects of financing frictions on growth. We show that financing frictions do

not need to be detrimental to innovation. In fact, constrained incumbents characterized

by low and volatile profits substitute production for innovation in the face of shrinking

cash reserves (and decreasing financial flexibility). Also, financing frictions increase the

entrants’ investment in innovation above the unconstrained level if their cash reserves are

sufficiently large (i.e., right after a financing round). In equilibrium, financing frictions

have a strong impact on industry composition and result in fewer but more valuable en-

trants. On aggregate, financing frictions importantly change the composition of growth by

deterring the entrants’ contribution to economic growth (the hurdle effect) but prompting

the incumbents’ contribution (the haven effect). If the haven effect dominates, financing

frictions can spur growth, which occurs if setup costs are sufficiently large.

36

A Appendix

A.1 The unconstrained economy

In this appendix, we report additional calculations and results referred to the uncon-strained economy described in section 3. The maximization problem of the consumptiongood sector (as reported in the main text) gives the demand curve for each frontier in-put, the price of each input (equation (7)), and the incumbents’ optimal production rate(equation (8)). Substituting (7) and (8) into (2), we obtain the aggregate output of theconsumption good sector:

Y∗t = (1− β)1β−2∫ 1

0

qjtdj ,

and the competitive labor wage W ∗t = β(1− β)1β−2 ∫ 1

0qjtdj .

V (t, q) denotes the value of an incumbent operating in the unconstrained economy (thesubscript j will be suppressed when it causes no confusion). The incumbent producesat the value-maximizing rate X̄∗t and pays out any operating profit to shareholders. Asimple calculation gives an expression for operating profits on each time interval:

β qt (p∗)1−

1β = βqt (1− β)

1β−1 .

Following standard arguments, V (t, q) satisfies the Hamilton-Jacobi-Bellman (HJB) equa-tion (9). To solve incumbents’ maximization problem, we conjecture incumbent value tobe linear in qt, V (t, q) = V (qt) = qtv

∗ for some v∗ > 0 representing scaled (time-invariant)incumbent value. Substituting into (9), we obtain

maxz∗

{µ∗ − (z

∗)2

2ζ + φz∗(λv∗ − v∗) + x∗d (ψκT + ψEκE − v∗)

}= r∗v∗, (37)

where the auxiliary quantity

µ∗ ≡ β (1− β)1β−1 (38)

represents scaled profits. The maximization of (37) delivers the optimal innovation ratein (11). Substituting the optimal innovation rate into (37) gives

µ∗ +φ2

2ζ(λ− 1)2(v∗)2 + x∗dψκT + x∗dψEκE = (x∗d + r∗)v∗,

which we solve with respect to v∗ (see the system of equation (42)).

Let us now consider entrant firms. Their value in the unconstrained economy isdenoted by V E(t, q̄). As explained in the main text, q̄t represents average quality offrontier inputs at time t. Following standard arguments, entrant value satisfies the HJBequation (12). To solve this maximization problem, we conjecture that the value of anactive entrant scales with q̄t, V

E(t, q̄) = q̄tv∗E, where v

∗E represents the scaled entrant

37

value. Substituting into (12) gives:

maxz∗E≥0

{− (z

∗E)

2

2ζE + φEz

∗E(Λv

∗ − v∗E − ΛκT )}

= (r∗ − g∗)v∗E, (39)

which we differentiate to obtain the entrants’ optimal innovation rate, which is given byequation (14). Substituting (14) into (39) gives

(r∗ − g∗)v∗E =φ2E2ζE

(Λv∗ − v∗E − ΛκT )2,

which we solve for v∗E :

v∗E = Λ(v∗ − κT ) +

r∗ − g∗ −√

(r∗ − g∗)2 + 2φ2EΛ(v∗ − κT )(r∗ − g∗)/ζEφ2E/ζE

. (40)

We plug the above expression into the free entry condition v∗E(m∗E) = κE. The free entry

condition, together with equation (14), implies that the inequality Λv∗ > κE +ΛκT needsto hold, otherwise the innovation rate would be negative. Also, the free entry conditionpins down the equilibrium measure of active entrants and the rate of creative destruction,see (1